About the Geometric Sequence Calculator
Unlike an arithmetic sequence that adds a fixed amount each step, a geometric sequence multiplies by a fixed ratio, producing the kind of accelerating growth or decay seen in compound interest, population models, and radioactive decay. This calculator finds any term and the running sum given a first term, common ratio, and term number.
How It Works
You provide the first term (a₁), the common ratio (r) that each term is multiplied by, and which term number (n) you want. The calculator returns the value of that nth term and the sum of every term from the first through the nth, along with a table listing up to the first ten terms.
Formula & Methodology
Each term in a geometric sequence is the previous term multiplied by the common ratio r, so the nth term is simply the first term multiplied by r raised to the power (n−1), since it takes n−1 multiplications to get from the first term to the nth. The sum formula comes from a telescoping trick: multiplying the whole sum by r and subtracting the original sum cancels every term except the first and last, leaving a compact closed-form expression. When r equals exactly 1, that formula would divide by zero, so the sum in that special case is just the first term multiplied by n, since every term in the sequence is identical.
Step-by-Step: Calculating It By Hand
- 1Identify the first term (a₁) and common ratio (r).
- 2To find the nth term, raise r to the power (n−1) and multiply by a₁.
- 3To find the sum of the first n terms, apply Sₙ = a₁(1 − rⁿ)/(1 − r), unless r = 1, in which case the sum is simply a₁ × n.
Examples
Growing sequence
With a₁ = 2 and r = 3, the 6th term is 2 × 3^5 = 486, and the sum of the first 6 terms is 728.
Shrinking sequence
With a₁ = 100 and r = 0.5, the terms head toward zero: 100, 50, 25, 12.5, and so on.
Advantages
- Instantly computes both a specific term and the running sum
- Displays a table of terms for a clear view of the pattern
- Correctly handles the special case where the common ratio equals 1
- Works for growth (r > 1) and decay (0 < r < 1) alike
Common Mistakes
- Confusing a geometric sequence (multiplying by a ratio) with an arithmetic sequence (adding a difference)
- Using n instead of n−1 as the exponent when finding the nth term
- Forgetting the sum formula breaks down when r equals exactly 1
- Not accounting for sign changes when the common ratio is negative
Edge Cases to Watch For
- When |r| is less than 1, the terms shrink toward zero rather than growing.
- A negative common ratio makes the sequence alternate in sign from term to term.
- If r equals exactly 1, every term is identical to the first, and the sum formula simplifies to a₁ × n.
- A term number below 1 is not valid, since sequence terms are indexed starting at 1.
Common Use Cases
- Modeling compound growth, such as investments or population increase
- Modeling exponential decay, such as radioactive half-life or depreciation
- Algebra and precalculus coursework on sequences and series
- Quick verification of hand-calculated geometric sequence problems